The upper traceable number of a graph

For a nontrivial connected graph G of order n and a linear ordering s: v ₁, v ₂, …, v _n of vertices of G, define . The traceable number t(G) of a graph G is t(G) = min{d(s)} and the upper traceable number t ⁺(G) of G is t ⁺(G) = max{d(s)}, where the minimum and maximum are taken over all linear orderings s of vertices of G. We study upper traceable numbers of several classes of graphs and the relationship between the traceable number and upper traceable number of a graph. All connected graphs G for which t ⁺(G) − t(G) = 1 are characterized and a formula for the upper traceable number of a tree is established. Research supported by Srinakharinwirot University, the Thailand Research Fund and the Commission on Higher Education, Thailand under the grant number MRG 5080075.     

Thermal properties and morphology of cassava starch grafted with different content of polystyrene

The interest in replacing synthetic polymers by biodegradable materials from renewable resources is steadily increasing. In this work, cassava starch grafted with different content of polystyrene (PS) was synthesized via free-radical polymerization using suspension polymerization technique. Thermal data of pure starch and the grafted starch with different content of PS were collected by simultaneous thermogravimetric (TG)–differential scanning calorimetry (DSC) setup in open alumina pans. Separately, typical DSC measurements were conducted in standard aluminum pans with lid. The data obtained by different methods are correlated and discussed. Morphology of cryogenic fracture surface of starch sample was studied by scanning electron microscopy. In order to obtain more reliable data about the processes taking place during the thermal treatment, the changes of surface morphology of starch treated at different temperatures are observed.     

On Connected Resolving Decompositions in Graphs

For an ordered k-decomposition of a connected graph G and an edge e of G, the -code of e is the k-tuple where d(e, G _i) is the distance from e to G _i. A decomposition is resolving if every two distinct edges of G have distinct -codes. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dim _d (G). A resolving decomposition of G is connected if each G _i is connected for 1 i k. The minimum k for which G has a connected resolving k-decomposition is its connected decomposition number cd(G). Thus 2 dim _d (G) cd(G) m for every connected graph G of size m 2. All nontrivial connected graphs of size m with connected decomposition number 2 or m have been characterized. We present characterizations for connected graphs of size m with connected decomposition number m – 1 or m – 2. It is shown that each pair s, t of rational numbers with 0 < s t 1, there is a connected graph G of size m such that dim _d (G)/m = s and cd(G)/m = t.     

Simultaneous Determination of Key Osmoregulants in Halophytes Using HPLC–ELSD

Osmoregulants are the substances produced by plants that assist in tolerating environmental stresses. Three commonly analysed osomoregulants include mannitol, betaine and proline. A simple, sensitive and rapid HPLC–ELSD method has been developed for the simultaneous analysis of these common osmoregulants in plant extracts. Osmoregulants were extracted using 80 % ethanol and separated on an NH₂ column using 0.1 % formic acid and acetonitrile as the mobile phase. Retention time repeatability was 0.85, 1.50, and 0.93 % for mannitol, betaine and proline, respectively. The limit of detection (μmol) was 1.43 × 10^?4, 7.81 × 10^?5 and 1.08 × 10^?4 for mannitol, betaine and proline, respectively. The developed method was applied to three different plant extracts, Stylosanthes guianensis, Atriplex cinerea and Rhagodia baccata. A second method using a C18 column with 0.1 % heptafluorobutyric acid and acetonitrile as the mobile phase proved to be a useful complementary method for verifying tentative peak identifications.     

Synergistic effect of nucleation and compatibilization on the polylactide and poly(butylene adipate-<Emphasis Type="Italic">co</Emphasis>-terephthalate) blend films

Polylactide (PLA) films blended with 10 wt% poly(butylene adipate-co-terephthalate) (PBAT) were prepared by using a twin screw extruder in the presence of the nucleating agent of titanium dioxide (TiO₂) and the compatibilizers of toluene diisocyanate (TDI) and PLA-grafted-maleic anhydride (PLA-g-MA). The synergistic effect of the nucleation and compatibilization on the properties and crystallization behavior of the PLA/PBAT (PLB) films was explored. The results showed that the addition of TiO₂ significantly enhanced the tensile strength and the impact tensile resistance of the PLB films while slightly decreased its thermal stability. In addition, the compatibilizers of TDI and PLA-g-MA in the system not only affected the crystallinity and cold crystallization process of the PLB films, but also increased the mechanical properties of them due to the improvement of the interfacial interaction between PLA and PBAT revealed by the morphological measurement. The synergistic effects of the nucleating agent and the compatibilizer afforded the blend films with increased tensile strength and impact tensile toughness, improved cold crystallization property and χ _c.     

Prediction of gas chromatographic peak width in capillary columns at different temperatures, carrier gas flows, column lengths, inside diameters and carbon numbers

Width factor (wf), relative band broadening (br) and retention factor are linearly correlated as ln wf = Inbr + ln k [Chromatographia (2002) 56, 99 -103]. The k and br are thermodynamic and kinetic energy parameters, and expanded to cover volumetric flow of carrier gas, temperature, phase ratio (beta) and carbon number (z). For columns of the same stationary phase and beta value but of different lengths, width at base (wb) can be predicted with the same numerical values of the constants. The average absolute differences between predicted and experimental wb for BP-1 columns of 30, 20 and 10 m are 1.27%, 1.21% and 1.37%, respectively. In predicting Wb of n-alkanes eluted from columns of the same stationary phase but of different ID, only the differences of ln beta are required for adjustment of one of the numerical values. The average absolute differences between Wb(cal) and Wb(exp) values for columns of 0.53, 0.32, 0.22 and 0.15 mm ID are 2.06, 3.67, 3.82 and 3.47%, respectively.     

Connected Resolvability of Graphs

For an ordered set W = {w ₁, w ₂,..., w _k} of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the k-vector r(v|W) = (d(v, w ₁), d(v, w ₂),... d(v, w _k)), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations with respect to W. A resolving set for G containing a minimum number of vertices is a basis for G. The dimension dim(G) is the number of vertices in a basis for G. A resolving set W of G is connected if the subgraph 〈W〉 induced by W is a nontrivial connected subgraph of G. The minimum cardinality of a connected resolving set in a graph G is its connected resolving number cr(G). Thus 1 ≤ dim(G) ≤ cr(G) ≤ n?1 for every connected graph G of order n ≥ 3. The connected resolving numbers of some well-known graphs are determined. It is shown that if G is a connected graph of order n ≥ 3, then cr(G) = n?1 if and only if G = K _n or G = K _1,n?1. It is also shown that for positive integers a, b with a ≤ b, there exists a connected graph G with dim(G) = a and cr(G) = b if and only if $\left( {a,b} \right) \notin \left\{ {\left( {1,k} \right):k = 1\;{\text{or}}\;k \geqslant 3} \right\}$ Several other realization results are present. The connected resolving numbers of the Cartesian products G × K ₂ for connected graphs G are studied.